Gauss Sum of Index 4: (2) Non–cyclic Case

Abstract

Assume that m ≥, p is a prime number, (m, p(p – 1)) = 1, −1 ∉ 〈p〉 ⊂ (ℤ/mℤ)* and [(ℤ/mℤ)* : 〈p〉] = 4. In this paper, we calculate the value of Gauss sum $G{\left( \chi \right)} = {\sum {_{{x \in \mathbb{F}^{ * }_{q} }} \chi {\left( x \right)}\zeta ^{{T{\left( x \right)}}}_{p} } }$ over ${\mathbb F}$ _q , where q = p ^f, $f = \frac{{\varphi {\left( m \right)}}} {4}$, χ is a multiplicative character of ${\mathbb F}$ _q and T is the trace map from ${\mathbb F}$ _q to ${\mathbb F}$ _p . Under our assumptions, G(χ) belongs to the decomposition field K of p in ℚ(ζ _m ) and K is an imaginary quartic abelian number field. When the Galois group Gal(K/ℚ) is cyclic, we have studied this cyclic case in another paper: "Gauss sums of index four: (1) cyclic case" (accepted by Acta Mathematica Sinica, 2003). In this paper we deal with the non–cyclic case.